## Tohoku Mathematical Journal

### Counting pseudo-holomorphic discs in Calabi-Yau 3-holds

Kenji Fukaya

#### Abstract

In this paper we define an invariant of a pair of a 6 dimensional symplectic manifold with vanishing 1st Chern class and its relatively spin Lagrangian submanifold with vanishing Maslov index. This invariant is a function on the set of the path connected components of bounding cochains (solutions of the $A_{\infty}$ version of the Maurer-Cartan equation of the filtered $A_{\infty}$ algebra associated to the Lagrangian submanifold). In the case when the Lagrangian submanifold is a rational homology sphere, it becomes a numerical invariant.

This invariant depends on the choice of almost complex structures. The way how it depends on the almost complex structures is described by a wall crossing formula which involves a moduli space of pseudo-holomorphic spheres.

#### Article information

Source
Tohoku Math. J. (2), Volume 63, Number 4 (2011), 697-727.

Dates
First available in Project Euclid: 6 January 2012

https://projecteuclid.org/euclid.tmj/1325886287

Digital Object Identifier
doi:10.2748/tmj/1325886287

Mathematical Reviews number (MathSciNet)
MR2872962

Zentralblatt MATH identifier
1244.57046

#### Citation

Fukaya, Kenji. Counting pseudo-holomorphic discs in Calabi-Yau 3-holds. Tohoku Math. J. (2) 63 (2011), no. 4, 697--727. doi:10.2748/tmj/1325886287. https://projecteuclid.org/euclid.tmj/1325886287

#### References

• S. Axelrod and I. Singer, Chern-Simons perturbation theory II, J. Differential Geom. 39 (1994) 173–213.
• S. Bosch, U. Güntzer and R. Remmert, Non-Archimedean analysis, A systematic approach to rigid analytic geometry, Grundlehren Math. Wiss. 261, Springer-Verlag, Berlin, 1984.
• C.-H. Cho, Strong homotopy inner product of an $A_{\infty}$-algebra, Int. Math. Res. Not. IMRN 13 (2008).
• K. Cieliebak, K. Fukaya and J. Latschev, Homological algebra related to Riemann surfaces with boundary, in preparation.
• K. Fukaya, Cyclic symmetry and adic convergence in Lagrangian Floer theory, Kyoto J. Math. 50 (2010), 521–591.
• K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian intersection Floer theory: Anomaly and obstructions (2006/7 version), preprint.
• K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian intersection Floer theory: Anomaly and obstructions, AMS/IP Stud. Adv. Math. 46, International Press, Amer. Math. Soc., Boston, Providence, 2009.
• K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory on compact toric manifolds I, Duke Math. J. 151 (2010), 23–174.
• K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Anti-symplectic involution and Floer cohomology, preprint, arXiv:0912.2646.
• K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariant, Topology 38 (1999), 933–1048.
• R. Gopakumar and C. Vafa, M-theory and topological strings-I,II, hep-th/9809187, hep-th/9812127.
• V. Iacovino, Open Gromov-Witten theory on Calabi-Yau three-folds I, preprint, arXiv:0907.5225v1.
• D. Joyce, Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds, Geom. Topol. 11 (2007), 667–725.
• D. Joyce, Kuranishi homology and Kuranishi cohomology, preprint, arXiv:0707.3572.
• H. Kajiura, Noncommutative homotopy algebra associated with open strings, Rev. Math. Phys. 19 (2007), 1–99.
• M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, preprint, arXiv:0811.2435.
• M. C. Liu, Moduli of $J$-holomorphic curves with Lagrangian boundary conditions and open Gromov-Witten invariants for an $S^1$-equivariant pair, preprint, math.SG/0210257 (2002).
• C. I. Lazaroiu, String field theory and brane superpotentials, J. High Energy Phys. 10 (2001), 829–862.
• R. Pandharipande, J. Solomon and J. Walcher, Disk enumeration on the quintic 3-fold, J. Amer. Math. Soc. 21 (2008), 1169–1209.
• J. Solomon, Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions, Thesis (Ph.D.)- Massachusetts Institute for Technology, 2006.
• R. P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on $K3$ fibrations, J. Differential Geom. 54 (2000), 367–438.
• A. Tomasiello, $A$-infinity structure and superpotentials, J. High Energy Phys. 9 (2001), 30.
• J. Walcher, Opening mirror symmetry on the quintic, Comm. Math. Phys. \text276 (2007), 671–689.
• J. Y. Welschinger, Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry, Invent. Math. 162 (2005), 195–234.